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Abstract:

Methods of producing layers of patterned graphene with smooth edges are
provided. The methods comprise the steps of fabricating a layer of
crystalline graphene on a surface, wherein the layer of crystalline
graphene has a crystallographically disordered edge, and decreasing the
crystallographic disorder of the edge of the layer of crystalline
graphene by heating the layer of crystalline graphene on the surface at
an elevated temperature in a catalytic environment comprising
carbon-containing molecules.

Claims:

1. A method for fabricating graphene structures, the method comprising:
(a) fabricating a layer of crystalline graphene on a surface, wherein the
layer of crystalline graphene has one or more crystallographically
disordered edges; and (b) subsequently decreasing the crystallographic
disorder of the one or more crystallographically disordered edges by
heating the layer of crystalline graphene on the surface at an elevated
temperature in a catalytic environment comprising carbon-containing
molecules, wherein the decrease in crystallographic disorder is catalyzed
by the surface.

2. The method of claim 1, wherein fabricating the layer of crystalline
graphene having one or more crystallographically disordered edges
comprises growing the layer of graphene on the metal surface and
lithographically patterning the layer of graphene.

3. The method of claim 1, wherein heating the layer of crystalline
graphene comprises heating the crystalline graphene to a temperature of
no greater than about 1000.degree. C.

4. The method of claim 1, wherein heating the layer of crystalline
graphene comprises heating the crystalline graphene to a temperature in
the range from about 700.degree. C. to about 900.degree. C.

5. The method of claim 1, wherein the carbon-containing molecules are in
the vapor phase.

7. The method of claim 1, wherein the method enlarges the area of the
layer of crystalline graphene.

8. The method of claim 2, wherein the metal comprises Cu.

9. The method of claim 2, wherein the layer of graphene is
lithographically patterned into an antidot lattice comprising holes
having irregularly shaped edges and further wherein the step of
decreasing the crystallographic disorder of the one or more
crystallographically disordered edges comprises converting the holes into
hexagonally-shaped holes and improving the alignment between the holes
and the hexagonal symmetry of the graphene lattice.

10. The method of claim 2, wherein the layer of graphene is
lithographically patterned into an array of nanoribbons having
irregularly shaped edges and further wherein the step of decreasing the
crystallographic disorder of the one or more crystallographically
disordered edges comprises improving the alignment between the edges of
the nanoribbons and the hexagonal symmetry of the graphene lattice.

11. Patterned graphene comprising a layer of graphene having plurality of
features defined therein, wherein the features have internal or external
edges that are aligned with the crystallographic direction of the
graphene lattice and further wherein the plurality of features comprises
at least 1,000 features.

12. The patterned graphene of claim 11, comprising a graphene antidot
lattice, wherein the plurality of features are a plurality of holes
defined in the layer of graphene, wherein the holes are hexagonal in
shape and aligned with the hexagonal crystal symmetry of the graphene
lattice, the graphene antidot lattice having an area of at least 1
μm2 and a hole density of at least 1.times.10.sup.11
holes/cm.sup.2.

13. The patterned graphene of claim 11, comprising a graphene nanoribbon
array, wherein the plurality of features are a plurality of graphene
nanoribbons aligned in a parallel arrangement along their longitudinal
axes, wherein the edges of the nanoribbons have an rms roughness no
greater than 5 nm and are aligned with the crystallographic direction of
the graphene, the graphene nanoribbon array comprising at least 1000
nanoribbons.

Description:

BACKGROUND

[0002] Graphene is a two-dimensional hexagonal network of sp2
hybridized carbon atoms. Graphene has been the subject of intense
research recently because of its outstanding electrical properties and
because of several intriguing phenomena that have been observed in the
two-dimensional carbon-based material. For many applications, a suitable
method for creating spatially defined patterns of appropriate size
(ranging from millimeter to nanometer scales) and shape in graphene is
necessary or desirable. For example, graphene can be transformed into a
semiconductor over large-areas by patterning it on the nanometer scale
into nanoribbons, quantum dots or continuous nanoperforated sheets
("antidot lattices"). Typically, this is accomplished by growing or
isolating a full graphene sheet and then etching away the unwanted
regions of the graphene from the top-down using lithography in
conjunction with reactive ion etching or through ion bombardment.

[0003] Unfortunately, top-down patterning creates graphene with
atomically-disordered edges, in the form of dangling bonds, defects,
chemical functionalization, and roughness. This edge disorder can degrade
graphene's electronic, optical, thermal and structural properties,
including its electron mobility and strength.

[0004] It has been reported that defective and disordered edges of
graphene will undergo reorganization at 1500-2000° C. via Joule
heating. (Jia et al., Science 2009, 323, 5922, 1701-1705.) However, the
high temperatures used during this process are too extreme for many
electronics applications.

SUMMARY

[0005] Methods for fabricating graphene structures having smooth edges are
provided. In some embodiments, the methods provide graphene structures
with features that are aligned with the crystallographic direction or
symmetry of the graphene in which they are defined. In one embodiment,
the method comprises fabricating a layer of crystalline graphene on a
surface, wherein the layer of crystalline graphene has one or more
crystallographically disordered edges; and subsequently decreasing the
crystallographic disorder of the one or more crystallographically
disordered edges by heating the layer of crystalline graphene on the
surface at an elevated temperature in a catalytic environment comprising
carbon-containing molecules.

[0006] The methods can be used to smooth the disordered edges in a layer
of graphene that result from top-down or bottom-up patterning of the
layer of graphene. For example, the methods can be used to smooth the
internal edges of holes in a graphene antidot lattice or to smooth the
edges of graphene nanoribbons in a graphene nanoribbon array.

[0007] In the methods, the step of heating the layer of crystalline
graphene can be carried out at temperatures of about 1000° C. or
lower. For example, the heating step can be carried out at temperatures
in the range from about 700° C. to about 900° C. Examples
of carbon-containing molecules that can be used during the heating step
include aliphatic hydrocarbons, aromatic hydrocarbons and derivatives
thereof.

[0008] Patterned graphene made in accordance with the present methods is
also provided. The patterned graphene comprises a layer of graphene
having a plurality of features defined therein, wherein the features have
internal or external edges that are aligned with the crystallographic
direction of the graphene lattice. The arrays can contain large numbers
of the features arranged in a regular or irregular pattern. For example,
the arrays can include at least 1,000, at least 10,000, at least 100,000,
or even at least 1,000,000 features in the layer of graphene. In some
embodiments, the graphene has a high density of features. For example, in
some embodiments the features have a density of at least 1×105
features/cm2 over an area of at least 1 μm2. This includes
embodiments in which the features have a density of at least
1×106, at least 1×107, at least 1×108,
at least 1×109 and at least 1×1010
features/cm2 over an area of at least 1 μm2.

[0009] An example of a graphene structure that can be fabricated using the
methods is a graphene antidot lattice comprising a layer of graphene in
which the features are holes arranged in a periodic array. Within the
antidot lattice the holes are hexagonal in shape and aligned with the
hexagonal crystal structure of the graphene lattice. Such graphene
antidot lattices can be fabricated over large areas with high hole
densities. For example, the present methods can be used to make graphene
antidot lattices with a hole density of at least 1×1011
holes/cm2 over an area of at least 1 μm2.

[0010] Another example of a graphene structure that can be fabricated
using the methods is a graphene nanoribbon array in which the features
are graphene nanoribbons aligned in a parallel arrangement along their
longitudinal axes, wherein the edges of the nanoribbons have an rms
roughness of 5 nm or less and are aligned with the crystallographic
direction of the graphene lattice. Such graphene nanoribbon arrays can
fabricated with a large number of nanoribbons. For example, in some
embodiments, the graphene nanoribbon array can comprise at least 1,000
nanoribbons.

[0011] Other principal features and advantages of the methods and
structures will become apparent to those skilled in the art upon review
of the following drawings, the detailed description, and the appended
claims.

BRIEF DESCRIPTION OF THE DRAWINGS

[0012] Illustrative embodiments of the invention will hereafter be
described with reference to the accompanying drawings.

[0014]FIG. 2 shows schematic illustrations of: (a) a layer of graphene
into which an array of circular holes with disordered edges has been
lithographically patterned, and (b) the layer of graphene after the
circular holes have been converted into hexagonal holes. The
corresponding scanning electron micrograph (SEM) images are shown in
panels (c) and (d).

DETAILED DESCRIPTION

[0015] Methods for forming layers of graphene with smooth and straight
edges are provided. Also provided are patterned graphene structures made
with the methods. The methods utilize a low-temperature edge annealing
process to catalyze the repair of random intrinsic defects along the
edges of a layer of graphene on a catalytic surface and to increase the
crystallographic alignment of features patterned into the graphene with
the crystallographic direction and symmetry of the graphene lattice. The
methods can be used to fabricate graphene nanoribbons with smooth,
straight edges that are aligned with the crystallographic direction of
the graphene lattice or to fabricate graphene antidot arrays with
hexagonal holes that are aligned with the hexagonal crystal structure of
the graphene lattice.

[0016] The methods can provide edges in the graphene that are ordered on
the sub-nanometer scale. This can be accomplished by reconstruction of
randomly spaced, oriented and/or shaped kinks, protrusions and
indentations along graphene edges which cause the edge structure to
deviate from a perfect crystallographic direction (e.g., zigzag or
armchair). FIG. 1 provides a schematic illustration of the conversion of
an atomically disordered graphene edge having randomly spaced, oriented
and shaped protrusions of carbon atoms (panel (a)) to an
atomically-smooth graphene edge having a perfect [1 1] zigzag
configuration (panel (b)). As shown in the figure, the carbon atoms at
the edge (dark spheres) may be terminated with hydrogen atoms (white
spheres).

[0017] In addition, the methods can decrease disorder by providing edges
that are straight on the sub-nanometer scale over considerable distances.
This can be accomplished by reconstructing the edges such that they are
better aligned with the crystallographic direction (e.g., zigzag or
armchair) of the graphene lattice, thereby eliminating the (typically
regularly spaced) bends or kinks that are characteristic of a lattice
misalignment.

[0018] The methods can also be used to decrease the crystallographic
disorder in a layer or layers of graphene that define a pattern of
features, such as a continuous layer of graphene into which an array of
holes has been patterned or a discontinuous layer of graphene into which
an array of graphene dots or ribbons has been patterned. The
crystallographic disorder in such system can be reduced by reconstructing
the irregularly shaped edges of the features such that they are smoother
and better aligned with the hexagonal crystal symmetry of the graphene
lattice.

[0019] The degree of edge smoothness can be characterized by the rms
roughness of the edge. Some embodiments of the methods produce edges with
an rms roughness of less than 5 nanometers. This includes embodiments in
which the edges have an rms roughness of less than 3 nanometers and
further includes embodiments in which the edges have an rms roughness of
less than 1 nanometer. The degree of edge straightness can be
characterized by the length over which the edge remains aligned with the
crystallographic lattice of the graphene, which is a function of the
angle of misalignment between the edge feature and the graphene lattice.
The lengths over which graphene edges can be rendered smooth and aligned
are considerable. For example, in some embodiments, the present methods
provide edges having a rms roughness of 5 nm or better and/or
crystallographic lattice alignment over lengths of at least 50 nm, at
least 100 nm, at least 1 μm, at least 100 μm or at least 1 mm.

[0020] The methods are based, at least in part, on the recognition that
the crystallographic edge structures of graphene are controlled by
thermodynamics, which seeks to reduce free energy and form ordered edges
along the crystalline directions in graphene, yet kinetic barriers to
reorganization normally prevent the edge carbon atoms from reorganizing
to achieve a lower energy state. In order to solve this problem, the
present methods use a substrate surface, to which the graphene is bonded,
that acts as a catalyst for the reorganization of edge carbon atoms. As a
result of the edge reorganization, the internal and/or external edges of
a nanostructured layer of graphene become smoother and more aligned with
the crystallographic lattice of the graphene. As a result of the methods,
the area of the layer of graphene can become smaller or larger, depending
on the nature of the reorganization.

[0021] One basic embodiment of the methods comprises the steps of
fabricating a layer of crystalline graphene on a catalytic surface,
wherein the layer of crystalline graphene has at least one
crystallographically disordered edge, and subsequently decreasing the
crystallographic disorder of the edge of the layer of crystalline
graphene by heating the layer of crystalline graphene on the catalytic
surface at an elevated temperature in a catalytic environment comprising
carbon-containing molecules, wherein the decrease in crystallographic
disorder is catalyzed by the catalytic surface.

[0022] As used herein, the phrase "a catalytic environment comprising
carbon-containing molecules" refers to an environment in which the
carbon-containing molecules are not only present, but also participate,
in some form, in the process of decreasing the crystallographic disorder.
The carbon-containing molecules can be in the form of vapor phase
molecules or can be in solid form as, for example, a carbon-containing
material (e.g., amorphous carbon or a carbon-containing polymer)
deposited on the surface. Without intending to be bound to any particular
theory of the catalytic mechanism, the inventors believe it is possible
that the catalytic surface may act as an intermediate stage for the
dynamic detachment and subsequent re-attachment of carbon atoms and
clusters to and from the edges of the nanostructured graphene. In this
mechanism, the carbon-containing molecules may suppress the sublimation
of carbon atoms from the catalytic substrate after they have broken away
from unstable bonding sites along the edge of the graphene, but before
they have diffused on the catalytic surface to stable bonding sites and
re-bonded to the edge. Alternatively, the surface may be catalyzing the
etching of disordered portions of the graphene edges by hydrogen present
in the fabrication chamber and the carbon-containing molecules may be
reconstructing the etched portions to form a smoother, more aligned edge
structure.

[0023] The initial crystallographic disorder in one or more edges of the
layer of graphene can be the result of imperfect bottom-up growth, as in
the case of chemical vapor deposition (CVD) grown graphene, or imperfect
top-down patterning, as in the case of lithographically patterned
graphene. Examples of lithographic techniques that can be used to form a
patterned layer of graphene (and which would be expected to create
crystallographically disordered edges) include, but are not limited to,
block copolymer lithography, nanosphere lithography, e-beam lithography,
interference lithography and photolithography. Methods for patterning
graphene layers using block copolymer lithography are described in U.S.
patent application publication number US 2011/0201201.

[0024] An advantage of the use of a catalytic surface is that edge
reorganization can be carried out at elevated temperatures (i.e.,
temperatures above ambient) that are substantially lower than those that
would be needed to carry out edge reorganization in the absence of the
catalytic surface. In addition, by allowing the edge reorganization to
proceed in the presence of an appropriate carbon-containing environment,
the edge reorganization can be conducted under conditions that promote
neither net shrinkage of the graphene via carbon sublimation nor
continued disordered growth of the graphene via vapor deposition.

[0025] The elevated temperature at which the graphene is annealed should
be high enough to induce edge reorganization on the catalytic substrate,
but is desirably still low enough to be compatible with the low
temperature processing conditions that are commonly used in electronic
device fabrication. Thus, in some embodiments, the edge reorganization is
carried out at a temperature of no greater than about 1200° C.
This includes embodiments in which the edge reorganization is carried out
at a temperature of no greater than about 1000° C. and further
includes embodiments in which the edge reorganization is carried out at a
temperature of no greater than about 900° C. For example, in
various embodiments, the edge reorganization can be carried out at
temperatures in the range from about 700° C. to about 1000°
C., including embodiments in which the edge reorganization is carried out
at temperatures in the range from about 800° C. to about
900° C. Temperatures in these ranges are substantially lower than
those that would be required to induce edge reorganization in the absence
of the catalytic surface. For example, in some embodiments annealing is
carried out at a temperature that is no greater than half the temperature
that would be required to induce edge reorganization in the absence of
the catalytic surface.

[0026] The carbon-containing molecules can be, for example, the types of
carbon-containing molecules that are used as carbon precursors in the CVD
growth of graphene. Such molecules include aliphatic and aromatic
hydrocarbons, such as methane, acetylene, ethylene, benzene and
derivatives thereof. These carbon-containing molecules can be mixed with
other, inert carrier gases, such as hydrogen and/or argon. The amount of
carbon-containing molecules present during the annealing process should
be sufficient either to suppress carbon sublimation or to replace at
least some of the carbon loss to sublimation during the edge
reorganization. The optimal amount will be depend upon variables such as
temperature, flow rates, catalyst materials, and type of
carbon-containing molecules used.

[0027] The catalytic surface upon which edge reorganization occurs can be
any surface upon which graphene can be grown that catalyzes the edge
reorganization. Such surfaces include metal and ceramic surfaces, such as
copper, nickel, ruthenium and silicon carbide surfaces.

[0028] A broad range of graphene structures can be made using the present
methods. These include structures wherein a layer or layers of graphene
have been patterned into an array of discontinuous features (e.g., arrays
of nanoribbons or nanodots) and further includes continuous graphene
layers into which arrays of features (e.g., holes) have been patterned.
The features in the arrays can be spaced with a regular periodicity or
can be randomly spaced in the graphene. In addition, the arrays may
themselves be formed from an arrangement of sub-arrays, as in the case
where a sub-array of nanoribbons is integrated into a field effect
transistor and a plurality of such sub-arrays are arranged on a wafer to
form an integrated circuit comprising many FETs over a large area. The
features in these arrays typically have at least one dimension (and
frequently two or all three dimensions) of 1000 nm or less. The sizes and
spacing of the features can be chosen to provide the graphene structures
with electronic and/or magnetic properties that are not present in the
unpatterned graphene.

[0029] The present methods can produce patterned graphene comprising a
layer of graphene having a large number of smooth-edged features
patterned therein. In some embodiments, the patterned graphene includes
such features at high densities over a large area. Because the present
methods do not substantially increase the size of lithographically
patterned features, and may actually decrease their size, the methods can
create smooth-edged features with dimension at least as small as the
disordered-edged features initially created by top-down or bottom-up
patterning techniques. In addition, because the present methods are based
on the repair of patterned graphene structures that can be made with high
throughput techniques, the methods can combine the ability to create
features having edges with sub-nanometer rms roughness with the ability
to fabricate a large number of features on a commercially feasible
timescale. Moreover, these features can be fabricated at high densities
over commercially practical areas (e.g., ≧1 μm2, ≧1
mm2, ≧1 cm2, or greater).

[0030] The features can be any structures that are defined by (e.g.,
patterned into) the graphene layer that are not inherently present as
part of the crystal structure of graphene. That is, the hexagonal shapes
that make up the 2D hexagonal crystal structure of graphene are not
features, as that term is used herein. Examples of features include holes
(of various shapes and sizes), strips (of various aspect ratios) and
dots.

[0031] One example of a graphene structure that can be made with the
present methods is a graphene antidot lattice. A graphene antidot lattice
comprises a layer or layers of graphene into which a periodic array of
holes has been patterned. The array of holes opens a bandgap in the
electronic structure of the graphene, which can provide it with useful
electronic and/or magnetic properties. The present methods are able to
produce antidot lattices wherein the holes are hexagonally-shaped and
aligned with the hexagonal crystal symmetry of the graphene. The use of
the present methods to fabricate a graphene antidot lattice is
illustrated in the Example, below.

[0032] The present methods can be used to fabricate antidot lattices
having hexagonal holes with diameters of 50 nm or less, 40 nm or less, 30
nm or less, 20 nm or less, or 10 nm or less. Arrays of these holes can be
formed at densities of, for example, at least 1×1011 (e.g., at
least 1×1012 or at least 5×1012) holes/cm2
over areas of at least 1 μm2. This includes high density antidot
lattices that extend over areas of at least 100 μm2, at least 1
mm2, and at least 10 mm2

[0033] Another example of a graphene structure that can be made with the
present methods is a graphene nanoribbon array. Graphene nanoribbons are
narrow strips (or "ribbons") of graphene having widths and
crystallographic edge structures that provide the ribbons with electronic
properties, such as electronic bandgaps, that are absent in larger area
layers of the graphene. Graphene nanoribbon arrays comprise a periodic
arrangement of a plurality of nanoribbons aligned along their
longitudinal axes. The present methods are able to produce graphene
nanoribbon arrays, wherein the longitudinal edges of the graphene
nanoribbons are smooth and aligned with the crystalline direction of the
graphene lattice.

[0034] The present methods can produce graphene nanoribbon arrays having a
high density of very narrow, smooth-edged, nanoribbons over a large area.
The width of the nanoribbons in the array is typically no greater than
about 10 nm and desirably no greater than about 5 nm. The lengths of the
nanoribbons are typically considerably greater than their widths. By way
of illustration only, the nanoribbons can have aspect ratios of 10:1,
50:1, 100:1, 1000:1, or greater. However, arrays of nanoribbons having
widths and lengths outside of these ranges can also be fabricated. The
spacing between the nanoribbons in the array (i.e., their `pitch`) can be
quite small in order to provide high density arrays. For example, in some
embodiments the graphene nanoribbon arrays will have a pitch of 500 nm or
less. This includes embodiments in which the pitch of the nanoribbons in
the array is 100 nm or less, further includes embodiments in which the
pitch is 50 nm or less and still further includes embodiments in which
the pitch is 10 nm or less. Arrays of these atomically-smooth nanoribbons
can include thousands of nanoribbons and can be formed over areas of, for
example, at least 1 μm2. This includes high density nanoribbon
arrays that extend over areas of at least 100 μm2, at least 1
mm2, and at least 10 mm2

Example

[0035] This example illustrates a method of producing atomically-smooth
hexagonal holes in a layer of graphene.

[0036] Materials and Methods

[0037] CVD Graphene Growth:

[0038] Large-area monolayer graphene was grown in a horizontal CVD furnace
with a 32 mm ID quartz tube. Copper foil (Alfa Aesar, product #13382) was
used as the growth catalyst and the catalytic surface for edge
reorganization. The foil was heated to 1050° C. under the flow of
340 sccm forming gas (95% argon, 5% hydrogen) and annealed under the same
conditions for 30 minutes. The furnace was then cooled to 1020°
C., where upon methane was introduced at 26 ppm and graphene was allowed
to grow for 16 hours. The foil was then rapidly cooled at
˜10° C./sec to 700° C. and then allowed to cool to
room-temperature.

[0040] To deposit a block copolymer film on the graphene/Cu foil, a
floating and transferring technique was used in which a thin film of the
diblock copolymer poly(styrene-block-methyl methacrylate) (P(S-b-MMA)) is
used as the starting material to form the BCP etch mask. P(S-b-MMA) forms
cylindrical domains in a hexagonal array. In order to ensure the lateral
phase segregation of the diblock copolymer into vertically oriented
cylinders on the graphene surfaces, two additional intermediate layers--a
layer of silicon oxide and a layer of random copolymer of styrene and
methyl methacrylate (P(S-r-MMA))--were used to provide a three-layered Si
oxide/P(S-r-MMA)/(P(S-b-MMA) film. This film was initially formed on a
separate "dummy" Si substrate and then transferred to the graphene/Cu
foil.

[0042] In order to release the three-layered film from the Si wafer and
float the film on an air-water interface, a 20% HF aqueous solution was
used to remove the silicon. The floated film on air-HF aqueous solution
interface was transferred to deionized (DI) water where it was picked up
by the surface of the graphene/Cu foil, and allowed to dry for one day.

[0043] The sample was then exposed to UV illumination (1000 mJ/cm2)
to selectively degrade the PMMA cylinders in the BCP layer of the film.
PMMA residue of those samples was removed by dipping in acetic acid for 2
minutes and rinsed with DI water. O2 plasma RIE (50 W, 10 mT, 10
sccm) was utilized to etch through the P(S-r-MMA), the silicon oxide, and
the graphene, providing a layer of graphene patterned with an array of
holes having atomically-disordered edges. After patterning, the remaining
etch mask was removed with ultrasonication in NMP and washing with
isopropanol.

[0044] Edge-Annealing:

[0045] The patterned layer of graphene was then annealed to alter the edge
structures of the holes in the graphene. This was done by reloading the
patterned graphene into the CVD chamber and heating the graphene to
850° C. under 50 sccm forming gas (95% argon, 5% hydrogen) and
0.068 sccm diluted methane (95% argon, 5% methane). The layer of graphene
was annealed at that condition for 55 minute, then was rapidly cooled
(10° C./sec) to 600° C. and, finally, cooled to room
temperature.

[0046] Results

[0047] Scanning electron microscope (SEM) images of the patterned graphene
prior to the edge anneal revealed that the holes appeared circular,
separated by substantially triangle-shaped vertices where three holes are
closest together. This geometry is represented schematically (panel (a))
and in the SEM image (panel (c)) of FIG. 2. After the edge anneal,
however, SEM images (FIG. 2(d)) revealed that the holes were converted
into hexagonal holes and the constriction widths (i.e., the widths of the
graphene material between the holes) became smaller and uniform, changing
the shape formed at the vertices of the holes, as shown in the schematic
representation in FIG. 2(b). This indicates that the carbons along the
internal edges of the holes have adopted a zigzag orientation, resulting
in atomically-smooth holes.

[0048] In this example, the BCP etch mask had multiple hexagonal domains
oriented randomly with respect to the lattice directions of the layer of
graphene. When the relative BCP domain lattice directions were compared,
it was observed that hexagon-shaped holes formed in those areas where the
BCP and the graphene has the same lattice direction, while circular
holes, having a lattice direction 30° off from the graphene
lattice direction, formed consistently in those areas where the BCP and
the graphene had different lattice directions. This indicated that when
the BCP and graphene lattices are oriented in the same direction, zigzag
edges are formed, creating hexagonal holes. However when the BCP and
graphene lattices are oriented in different directions, some zigzag and
some armchair edges are formed, creating circular holes.

[0049] The word "illustrative" is used herein to mean serving as an
example, instance, or illustration. Any aspect or design described herein
as "illustrative" is not necessarily to be construed as preferred or
advantageous over other aspects or designs. Further, for the purposes of
this disclosure and unless otherwise specified, "a" or "an" means "one or
more". Still further, the use of "and" or "or" is intended to include
"and/or" unless specifically indicated otherwise.

[0050] The foregoing description of illustrative embodiments of the
invention has been presented for purposes of illustration and of
description. It is not intended to be exhaustive or to limit the
invention to the precise form disclosed, and modifications and variations
are possible in light of the above teachings or may be acquired from
practice of the invention. The embodiments were chosen and described in
order to explain the principles of the invention and as practical
applications of the invention to enable one skilled in the art to utilize
the invention in various embodiments and with various modifications as
suited to the particular use contemplated. It is intended that the scope
of the invention be defined by the claims appended hereto and their
equivalents.